Euler’s Theorem Denialism

The U.S. Department of Energy employs physics Ph.D.s to manage our nuclear weapons. How would you feel if some of them wrote blog posts saying that it is possible to build a perpetual motion machine? What if they did this to signal their loyalty to some club of physicists? Wouldn’t you wonder why membership in this club was important enough get them say that they do not believe the second law of thermodynamics? And what kind of physics club would use an endorsement of the perpetual motion machine as a loyalty oath?

In a recent post, David Andolfatto, who is a Vice President at the Federal Reserve Bank of St. Louis, gives a brazen display of mathiness–brazen because he denies Euler’s theorem, which for economists is about the same as denying the second law of thermodynamics is for physicists.

The type of mathiness that is hardest to root out is the opaque mathiness illustrated by Lucas (2009). It combines math that is hard to understand with verbal claims that can be shown to be misleading, but only after a careful analysis of the math. By taking advantage of ambiguity and misdirection, its verbal claims can mislead without saying anything that is actually false.

Andolfatto’s brazen mathiness involves a verbal statement about a mathematical model that flies in the face of an impossibility theorem. No model can do what he claims his does. No model can have a competitive equilibrium with price-taking behavior and partially excludable nonrival goods.

If you are not an economist, this would be a model in which someone who has a monopoly on an idea can charge for its use, but somehow is unable to influence the price that users have to pay, which should sound implausible at least. If you are an economist, you know that there is a very simple argument based on Euler’s theorem that proves this type of model is impossible. The proof goes back a long way. I know that Karl Shell invoked it in the late 1960s. I restated it in the AER article of mine that Andolfatto quotes, so it was fresh in his memory. Dietz Vollrath has a recent post that works through the logic again.

In its most general form, the proof relies on a step that invokes a fact about production processes: If you double all the rival inputs (the inputs you can touch or stub your toe on) you double the output. Some economists try to evade the theorem by denying the possibility of replication. But Andolfatto’s paper makes the required assumption about production openly–constant returns to scale in rival inputs. So he’s got no wriggle room. I can’t for the life of me see how Andolfatto thinks he can evade Euler’s theorem.

It is certainly possible that he is confused. But if you were confused, wouldn’t you try to understand the proof that says what you want to claim has to be false before you go ahead and claim it anyway?

So the only conjecture that makes sense to me is that this is part of being in the club. But if so, there is an interesting differentiation of roles by status. Lucas knows a proof when he sees one and is careful to avoid getting caught saying something that is provably false. He sticks to the Marshallian framework where nonrival goods are nonexcludable. Prescott, in his paper with McGrattan (AER 2010) least tries to cover his tracks by inserting his imaginary input, location. Then he can simply assert that it does not get compensation and he can take the income that Euler’s Theorem allocates to location and transfer it to his nonrival inputs. From Minnesotta types who are so preachy about avoiding free parameters, you have to admire the audacity of adding an entirely free variable. But Prescott, like Lucas, he is careful not to deny Euler’s Theorem.

But the next level down in the hierarchy, followers like Boldrin and Levine are willing to just embarrass themselves:

It is widely believed that competitive equilibrium always results in prices equal to marginal cost. Hence the belief that competition is inconsistent with innovation. However widespread this belief may be, it is not correct. It is true only in the absence of capacity constraints, …(2008, p. 436)

They say they can ignore Euler’s Theorem, because in their bizarro version of a competitive equilibrium, prices for inputs do not equal marginal products.

But instead of presenting a competitive equilibrium of this type, they present a model* with an innovator who turns out to have market power. Their solution? The innovator has to be a price taker because they say so:

Making the initial single innovator behave competitively even in the very first period may be a source of misunderstanding. Since, by necessity, she has a monopoly in the initial period, why do we not take account of her incentive to restrict the initial supply…? (Boldrin and Levine, 2008, p. 438)

So in the analogy from physics, Boldrin and Levine say that it is possible to build a perpetual motion machine, but to their credit, at the last moment they back off and invoke the can-opener joke: “Well we haven’t actually built a perpetual motion machine, … so let’s just assume that we have one.” So what to make of Andolfatto, who goes all in: “I built one back in 1999 when I was in graduate school.”

It is as if the prophets who lead the Mormon Church never have to say anything specific about where the Book of Mormon comes from. Officials at the next level down have to say that Joseph Smith transcribed it from golden plates that he found in upstate New York. Regular church members have to say that it is a literal fact that the gold plates were written in 400 AD by Mormon and Moroni, using a script called reformed Egyptian, and were held together in a D-ring binder.

The provably false statements that economists like Andolfatto make (and he is certainly not alone) may be more than mere signals. They might be an irreversible commitment to stay at an institution where his club is already in control because they prevents someone from ever being employable at a competitive institution where logic is still valued.

Having a strong affiliation with other members of a religion can be a good thing, so I would not be bothered to learn that some of our nuclear physicists are Mormons. But I would be worried to learn that some of them were members of the perpetual motion club. And I am worried that some outposts of the Fed have been permanently colonized by economists who are on the record as Euler’s-Theorem-deniers.


* Thx to Jag Bhalla (@hangingnoodles) for catching the missing words.